The present invention relates to a computer-implemented method of calculating various types of characteristic curves of a centrifugal fluid machine (pump or the like), and a computer-readable storage medium having a program recorded thereon for calculating various types of characteristic curves of a centrifugal fluid machine. The present invention also relates to a computer-implemented method of geometrically converting coordinates in drawing a high-order curve, and a computer-readable storage medium having a program recorded thereon for geometrically converting coordinates in drawing a high-order curve.
When customers have requested a pump having a prescribed performance (desired flow rate and head), the following method has heretofore been employed to supply a pump that meets the desired performance.
First, a pump capable of providing the requested performance (flow rate and head) is selected from among numerous types of pumps. Specifically, as shown in FIG. 6, a pump is selected to have such characteristics that coordinates A1 which are determined by the requested flow rate and head are located between a flow-head characteristic curve (Q-H characteristic curve) Y1 with an impeller having a diameter of 100 mm and a flow-head characteristic curve (Q-H characteristic curve) Y2 with an impeller having a diameter of 50 mm, or half the size, in the cases where parts other than an impeller housed in a pump casing are not changed, but the impeller is changed only in diameter. In other words, the Q-H characteristic curves Y1 and Y2 are calculated for a plurality of types of pumps in advance, and pumps having pump characteristics which are located between the curves Y1 and Y2 are selected from the plurality of types of pumps.
It is possible to obtain a pump with the required flow rate by setting the diameter of the impeller of the selected pump to 100 mm and throttling the opening of the valve mounted on the discharge port of the pump to raise the head against the flow rate on the Q-H characteristic curve Y1.
However, since an unnecessary increase in head is caused by throttling the opening of the valve in this method, loss of the motor power or the like is increased, and hence the running cost is problematically increased due to the increase in electric power consumption.
In order to solve the above problems, there has been proposed a method of selecting an impeller having such a diameter that a Q-H characteristic curve passes through the requested flow rate and head, rather than a method of simply setting the diameter of the impeller housed in the pump casing to 100 mm.
The following method is employed to select such an impeller, for example. In FIG. 7, a Q-H characteristic curve Y3 which is located intermediately between the Q-H characteristic curves Y1 and Y2 is calculated, and then it is determined whether the curve Y3 passes through the coordinates A1 for the requested flow rate and head. When the coordinates A1 are larger than the Q-H characteristic curve Y3, a Q-H characteristic curve Y4 which is located intermediately between the Q-H characteristic curves Y1 and Y3 is calculated, and then it is determined whether the curve Y4 passes through the coordinates A1. This process of calculations is repeated until a Q-H curve passing through the coordinates A1 is found. Based on the Q-H characteristic curve passing through the coordinates A1 that is found above, the diameter of the impeller is calculated, and a pump incorporating the impeller having the calculated diameter is provided to the customer.
The following method has heretofore employed to calculate the Q-H characteristic curve Y3 located intermediately between the two Q-H characteristic curves Y1 and Y2 in FIG. 7, based on these two Q-H characteristic curves Y1 and Y2. As shown in FIG. 8, this method employs two Q-H characteristic curves YH1 and YH2 and Q-E characteristic curves (flow-efficiency characteristic curves) YE1 and YE2 which correspond to the Q-H characteristic curves YH1 and YH2, respectively. The flow rates are calculated at a plurality of points on the Q-E characteristic curves YE1 and YE2 which have the same efficiency, including P11 and P21, P12 and P22, P13 and P23, P14 and P24, P15 and P25, and P16 and P26. The heads corresponding to the respective flow rates are then calculated. Although P16 and P26 are the best efficiency points, respectively, and do not have the same efficiency, they are assumed to have the same efficiency in this example.
For example, with regard to the points P11 and P21, flow rates Q11 and Q21 which correspond to an efficiency ER1 are calculated at the points P11 and P21. However, it is not easy to calculate the flow rates Q11 and Q21 on the X-axis from the efficiency ER1 on the Y-axis in a high-order curve, but many calculations are required. Moreover, since the number of points at which the flow rates are calculated is 12 in this example, the similar calculations should be performed 12 times.
Next, the flow rates Q11 and Q21 are substituted for the two Q-H characteristic curves YH1=fH1(x) and YH2=fH2(x) to calculate the respective heads H11 and H21 which correspond to the flow rates Q11 and Q21 calculated above. The other heads are also calculated in the similar manner.
A coordinate point R1 (QR1, HR1) is estimated with the following equations from the calculated flow rates Q11 and Q21 and the calculated heads H11 and H21. The other coordinate points R2-R6 are also calculated in the similar manner.
HR1={(H11xe2x88x92H21)/2}+H21
QR1={(Q11xe2x88x92Q21)/2}+Q21
A new Q-H characteristic curve YH3 is then calculated by the least-square approximation of the sequence of the calculated coordinate points R1-R6.
Next, because a coordinate point S1 (QR1, ER1) on a Q-E characteristic curve YE3 which corresponds to the calculated Q-H characteristic curve YH3 has been calculated in the above calculation, the Q-E characteristic curve YE3 is calculated by the least-square approximation of the sequence of the calculated coordinate points S1-S6.
Complicated and massive calculations are required to derive a high-order equation with the least-square method. Since such calculations should be performed for deriving two high-order equations for the Q-H characteristic curve YH3 and the Q-E characteristic curve YE3, further massive calculations are required.
Then, it is determined whether the Q-H characteristic curve YH3 calculated with the above method passes through the coordinates A1 for the requested flow rate and head as described with reference to FIG. 7. If the Q-H characteristic curve YH3 does not pass through the coordinates A1, the above calculation is repeated.
Assuming that the calculations for calculating the Q-H characteristic curve and the Q-E characteristic curve are repeated five times, for example, a value on the X-axis should be calculated from a value on the Y-axis in the high-order equation 60 times, and the least-square approximation should be performed 10 times. Therefore, it is necessary to perform massive and complicated calculations, which cannot be performed on a personal computer at a practical speed but requires a host computer.
The performance curve for a pump, such as the Q-H characteristic curve described above, is usually expressed by representing the flow rate as [m3/min] on the horizontal axis and the head as [m] on the vertical axis. While this system of units (coordinates) is usually used in Japan, the Q-H characteristic curve should be displayed with another system of units (coordinates) of another country in the case of selling products in that country, for example. Specifically, it may be necessary to display a Q-H characteristic curve in [USG(US gallon)/min] on the horizontal axis and in [feet] on the vertical axis, for example.
The following method has heretofore been employed to convert a characteristic curve expressed in a prescribed system of units (coordinates) into a characteristic curve (high-order curve) expressed in a different system of units (coordinates) by conversion of the units for drawing the converted characteristic curve with a computer. First, values of a plurality of points (x, y) on the characteristic curve which is formed from a high-order equation expressed in the prescribed system of units (coordinates) are calculated. Next, these values are converted into values of a plurality of points (x, y) in a different desired system of units (coordinates) by conversion of the units. The coefficients for each of orders in the high-order equation passing through the plurality of calculated points are calculated by the least-square approximation using the least-square method. The results are drawn as a characteristic curve converted into the desired units.
However, as described above, complicated and massive calculations are required to derive the high-order equation by the least-square method, and hence such calculations take a large amount of time even with use of a computer. Further, the calculated characteristic curve is not necessarily accurate.
The present invention has been made in view of the above drawbacks. It is therefore a first object of the present invention to provide a computer-implemented method of calculating various types of characteristic curves of a centrifugal fluid machine and a computer-readable storage medium having a program recorded thereon for calculating various types of characteristic curves of a centrifugal fluid machine which can easily calculate a Q-H characteristic curve, a Q-E characteristic curve, a Q-NPSH characteristic curve, or the like.
A second object of the present invention is to provide a computer-implemented method of geometrically converting coordinates in drawing a high-order curve and a computer-readable storage medium having a program recorded thereon for geometrically converting coordinates in drawing a high-order curve which can reduce time required for calculations and can obtain an accurate high-order curve (performance curve).
In order to attain the first object, according to the present invention, there is provided a computer-implemented method of calculating various types of characteristic curves of a centrifugal fluid machine, wherein two prescribed characteristic curves Y1=a11+a12x+a13x2+ . . . +a1nx(nxe2x88x921) and Y2=a21+a22x+a23x2+ . . . +a2nx(nxe2x88x921) formed of high-order equations for a centrifugal fluid machine are used to calculate a characteristic curve Y3=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) formed of a high-order equation which passes through different coordinates (x3, y3), the method characterized by comprising: selecting prescribed coordinates (x1, y1) on the characteristic curve Y1 and corresponding prescribed coordinates (x2, y2) on the characteristic curve Y2; and calculating and outputting a characteristic curve Y3=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) formed of a high-order equation which passes through different coordinates (x3, y3), with use of an equation bn={a1nkh1(1/kq1)(nxe2x88x921)xc3x97(y3xe2x88x92y2)/(y1xe2x88x92y2)}+{a2nkh2(1/kq2)(nxe2x88x921)xc3x97(y1xe2x88x92y3)/(y1xe2x88x92y2)}, wherein kq1 is a ratio (=x3/x1) of the selected coordinate x1 and the different coordinate x3, kh1 is a ratio (=y3/y1) of the selected coordinate y1 and the different coordinate y3, kq2 is a ratio (=x3/x2) of the selected coordinate x2 and the different coordinate x3, and kh2 is a ratio (=y3/y2) of the selected coordinate y2 and the different coordinate y3.
The characteristic curves Y1, Y2, and Y3 can be applied to flow-head characteristic curves, flow-efficiency characteristic curves, or flow-net positive suction head characteristic curves. When a flow-head characteristic curve is selected, for example, the required characteristic curve is calculated as described below.
Specifically, a computer-implemented method of calculating a flow-head characteristic curve of a centrifugal fluid machine uses two prescribed flow-head characteristic curves Y1=a11+a12x+a13x2+ . . . +a1nx(nxe2x88x921) and Y2=a21+a22x+a23x2+ . . . +a2nx(nxe2x88x921) formed of high-order equations for a centrifugal fluid machine to calculate a flow-head characteristic curve Y3=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) formed of a high-order equation which passes within permissible values for an inputted different flow rate Qr and head Hr; selects a head H1 for a flow rate Q1 at the best efficiency point on the flow-head characteristic curve Y1, a head H2 for a flow rate Q2 at the best efficiency point on the flow-head characteristic curve Y2, and a head H3 for a flow rate Q3 at the best efficiency point on a desired provisional flow-head characteristic curve Y3=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) formed of a high-order equation; and calculates a new flow-head characteristic curve Y3 with use of an equation bn={a1nkh1(1/kq1)(nxe2x88x921)xc3x97(H3xe2x88x92H2)/(H1xe2x88x92H2)}+{a2nkh2(1/kq2)(nxe2x88x921)xc3x97(H1xe2x88x92H3)/(H1xe2x88x92H2)}, and outputs the flow-head characteristic curve Y3 when the flow-head characteristic curve Y3 passes within permissible values for the inputted different flow rate Qr and head Hr, and otherwise corrects respective coefficients of the equation Y3=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) and recalculates a head H3 for the flow rate Q3 at the best efficiency point on the flow-head characteristic curve Y3 using the corrected coefficients, wherein kq1 is a ratio (=Q3/Q1) of the selected flow rates Q1 and Q3, kh1 is a ratio (=H3/H1) of the selected heads H1 and H3, kq2 is a ratio (=Q3/Q2) of the selected flow rates Q2 and Q3, and kh2 is a ratio (=H3/H2) of the selected heads H2 and H3.
Further, according to the present invention, there is provided a computer-readable storage medium having a program recorded thereon for executing a procedure with a computer, the procedure comprising: selecting prescribed coordinates (x1, y1) on a characteristic curve Y1 and corresponding prescribed coordinates (x2, y2) on a characteristic curve Y2 using the two prescribed characteristic curves Y1=a11+a12x+a13x2+ . . . +a1nx(nxe2x88x921) and Y2=a21+a22x+a23x2+ . . . +a2nx(nxe2x88x921) formed of high-order equations for a centrifugal fluid machine; selecting prescribed coordinates (x3, y3) on a characteristic curve Y3 formed of a high-order equation indicating a desired equation Y3=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921); and calculating and outputting a characteristic curve Y3=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) formed of a high-order equation which passes through the coordinates (x3, y3), with use of an equation bn={a1nkh1(1/kq1)(nxe2x88x921)xc3x97(y3xe2x88x92y2)/(y1xe2x88x92y2)}+{a2nkh2(1/kq2)(nxe2x88x921)xc3x97(y1xe2x88x92y3)/(y1xe2x88x92y2)}, wherein kq1 is a ratio (=x3/x1) of the selected coordinates x1 and x3, kh1 is a ratio (=y3/y1) of the selected coordinates y1 and y3, kq2 is a ratio (=x3/x2) of the selected coordinates x2 and x3, and kh2 is a ratio (=y3/y2) of the selected coordinates y2 and y3.
According to the present invention, by direct X-Y coordinate transformation of a flow-head characteristic curve of a high-order equation, a flow-head characteristic curve of a different high-order equation can easily be calculated, and hence it is not necessary to calculate the X coordinate from the Y coordinate as in the conventional example. Further, it is not necessary to calculate a high-order equation by the least-square method, thereby enabling practical and fast calculations at a processing speed suitable for a personal computer.
In order to attain the second object, according to the present invention, there is provided a computer-implemented method of converting coordinates in drawing a high-order curve, wherein a high-order curve Y1=a1+a2x+a3x2+ . . . +anx(nxe2x88x921) expressed in prescribed coordinates is converted into a high-order curve Y2=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) expressed in different coordinates for drawing the converted high-order curve with a computer, the method characterized by comprising: calculating a geometric conversion coefficient kx (=a value of the different coordinate/a value of the prescribed coordinate) for the direction of the X coordinate axis and a geometric conversion coefficient ky (=a value of a different coordinate/a value of a prescribed coordinate) for the direction of the Y coordinate axis; and calculating respective coefficients bn (n=1xe2x88x92n) of the equation Y2=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) according to an equation bn=anxc3x97ky/(kx)(nxe2x88x921) with use of the coefficients an (n=1xe2x88x92n) for each of orders of the high-order curve Y1 and the geometric conversion coefficients kx and ky, and substituting the coefficients bn for the equation Y2=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) to convert the high-order curve Y1 into the high-order curve Y2.
Further, according to the present invention, there is provided a computer-readable storage medium having a program recorded thereon for executing a procedure with a computer, the procedure comprising: calculating a geometric conversion coefficient kx (=a value of a different coordinate/a value of a prescribed coordinate) for the direction of the X coordinate axis and a geometric conversion coefficient ky (=a value of the different coordinate/a value of the prescribed coordinate) for the direction of the Y coordinate axis for converting between a high-order curve Y1=a1+a2x+a3x2+ . . . +anx(nxe2x88x921) expressed in the prescribed coordinates and a high-order curve Y2=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) expressed in the different coordinates; calculating respective coefficients bn (n=1xe2x88x92n) of the equation Y2=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) according to an equation bn=anxc3x97ky/(kx)(nxe2x88x921) with use of the coefficients an (n=1xe2x88x92n) for each of orders of the high-order curve Y1 and the geometric conversion coefficients kx and ky, and substituting the coefficients for the equation Y2=b1+b2x+b3x2+ . . . +bnx(nxe2x88x921) to convert the high-order curve Y1 into the high-order curve Y2; and drawing the converted high-order curve Y2.
According to the present invention, coordinates of a high-order curve can geometrically be converted simply by converting respective coefficients each of orders of a function. Moreover, the calculated performance curve is accurate.